Isoparametric hypersurfaces in spheres, integrable nondiagonalizable systems of hydrodynamic type, and N-wave systems

AbstractIsoparametric hypersurface Mn ⊂Sn + 1 can be defined as an intersection of the unit sphere r2 = (u1)2 + … + (un + 2)2 = 1 with the level set F(u) = const of a homogeneous polynomial F of degree g, satisfying Cartan-Munzner equations (▽F)2 = g2r2g − 2, Δ F = crg − 2 c = const. We introduce a hamiltonian system of hydrodynamic type uit = 1gδijddx∂F∂uj, with the hamiltonian operator δijddx and the hamiltonian density F(u)g. Under the additional assumption of the homogeneity of the hypersurface Mn, the restriction of this system to Mn proves to be nondiagonalizable, but integrable and can be transformed to an appropriate integrable reduction of the N-wave system. Possible generalizations to isoparametric submanifolds (finite or infinite dimensional) are also briefly indicated